Abstract
In the N-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian N-body problem with equal masses, N ≧ 3, there are at least 2N-3 + 2[(N-3)/2] different main simple choreographies. This confirms a conjecture given by Chenciner et al. (Geometry, mechanics, and dynamics. Springer, New York, pp 287–308, 2002). All the simple choreoagraphies we prove belong to the linear chain family.
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References
Barutello V., Terracini S.: Action minimizing orbits in the n-body problem with simple choreography constraint. Nonlinearity 17(6), 2015–2039 (2004)
Chen K.-C.: Action-minimizing orbits in the parallelogram four-body problem with equal masses. Arch. Ration. Mech. Anal., 158(4), 293–318 (2001)
Chen K.-C.: Binary decompositions for planar N-body problems and symmetric periodic solutions. Arch. Ration. Mech. Anal., 170(3), 247–276 (2003)
Chen, K.-C.: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Ann. Math. (2), 167(2), 325–348, 2008
Chen, K.-C.: Keplerian action functional, convex optimization, and an application to the four-body problem, 2013
Chen K.-C., Lin Y.-C.: On action-minimizing retrograde and prograde orbits of the three-body problem. Commun. Math. Phys., 291(2), 403–441 (2009)
Chen K.-C., Ouyang T., Xia Z.: Action-minimizing periodic and quasi-periodic solutions in the n-body problem. Math. Res. Lett., 19(2), 483–497 (2012)
Chenciner, A.: Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002). Higher Ed. Press, Beijing, 279–294, 2002
Chenciner, A.: Are there perverse choreographies? New Advances in Celestial Mechanics and Hamiltonian Systems, 63–76. Kluwer/Plenum, New York, 2004
Chenciner, A.: Symmetries and “simple” solutions of the classical n-body problem. XIVth International Congress on Mathematical Physics, pages 4–20. World Sci. Publ., Hackensack, NJ, 2005
Chenciner A.: Féjoz J.: Unchained polygons and the N-body problem. Regul. Chaotic Dyn., 14(1), 64–115 (2009)
Chenciner, A., Gerver, J., Montgomery, R., Simó, C.: Simple choreographic motions of N bodies: a preliminary study. Geometry, mechanics, and dynamics. Springer, New York, 287–308, 2002
Chenciner A., Montgomery R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. (2) 152(3), 881–901 (2000)
Ferrario D.L., Terracini S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math., 155(2), 305–362 (2004)
Fujiwara T., Montgomery R.: Convexity of the figure eight solution to the three-body problem. Pac. J. Math., 219(2), 271–283 (2005)
Fusco G., Gronchi G.F., Negrini P.: Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem. Invent. Math., 185(2), 283–332 (2011)
Gordon W. B.: A minimizing property of Keplerian orbits. Amer. J. Math., 99(5), 961–971 (1977)
Kapela T., Zgliczyński P.: The existence of simple choreographies for the N-body problem—a computer-assisted proof. Nonlinearity 16(6), 1899–1918 (2003)
Marchal, C.: How the method of minimization of action avoids singularities. Celest. Mech. Dynam. Astronom., 83(1-4), 325–353, 2002. Modern celestial mechanics: from theory to applications (Rome, 2001)
Montaldi, J., Steckles, K.: Classification of symmetry groups for planar n-body choreographies. Forum Math. Sigma, 1:e5, 55, 2013
Montgomery, R.: Figure 8s with three bodies. count.ucsc.edu/rmont/papers/fig8total.pdf.
Montgomery R.: The N-body problem, the braid group, and action-minimizing periodic solutions. Nonlinearity 11(2), 363–376 (1998)
Moore C.: Braids in classical dynamics. Phys. Rev. Lett., 70(24), 3675–3679 (1993)
Palais R.S.: The principle of symmetric criticality. Commun. Math. Phys., 69(1), 19–30 (1979)
Shibayama M.: Variational proof of the existence of the super-eight orbit in the four-body problem. Arch. Ration. Mech. Anal., 214(1), 77–98 (2014)
Simó, C.: New families of solutions in N-body problems. In European Congress of Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math. Birkhäuser, Basel, 101–115, 2001
Venturelli, A.: Application de la Minimisation de L’action au Problème des N Corps dans le plan et dans L’espace,. PhD thesis, Université Denis Diderot in Paris, 2002
Yu G.: Periodic Solutions of the Planar N-Center Problem with topological constraints. Discrete Contin. Dyn. Syst., 36(9), 5131–5162 (2016)
Yu, G.: Shape Space Figure-8 Solution of Three Body Problem with Two Equal Masses. accepted by Nonlinearity, arXiv:1507.02892, 2016
Yu, G.: Spatial double choreographies of the Newtonian 2n-body problem. arXiv:1608.07956, 2016
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Communicated by P. Rabinowitz
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Yu, G. Simple Choreographies of the Planar Newtonian N-Body Problem. Arch Rational Mech Anal 225, 901–935 (2017). https://doi.org/10.1007/s00205-017-1116-1
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DOI: https://doi.org/10.1007/s00205-017-1116-1